Linear functional equations involving Babbage’s equation

Functional equations arise in many branches of mathematics, for example, dynamical systems [1, 19, 24, 43], functional analysis [42], geometry [8, 9], information theory [3], wavelet theory [20, 21], and special functions [27]. They also occur in other disciplines such as physics [22, 33], engineering [15, 16], economics [4, 23] and so on. ∗Supported by the Fund of the Sichuan Provincial Education Department (13ZB0005), NSFC # 11101295 and the Key Project of the Sichuan Provincial Department of Education (12ZA086).